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Continuous Models
Chapter 4
Bacteria Growth-Revisited
• Consider bacteria growing in
a nutrient rich medium
• Variables
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– Time, t
– N(t) = bacteria density at time t
• Dimension of N(t) is # cells/vol.
• Parameters
– k = growth/reproduction rate
per unit time
• Dimension of k is 1/time
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Bacteria Growth Revisited
• Now suppose that bacteria densities are
observed at two closely spaced time
points, say t and t + t
• If death is negligible, the following
statement of balance holds:
Bacteria
Density @
N(t+t)
t +t
=
Bacteria
density @
time t
+
=
N(t)
+
New bacteria
Produced in the
Interval t + t - t
kN(t) t
Bacteria Growth Revisited
• Rearrange these terms
• Assumptions
N(t  t)  N(t)
 kN
t
– N(t) is large--addition of one or several new cells
is of little consequence
– There 
is no new mass generated at distinct
intervals of time, ie cell growth and reproduction is
not correlated.
• Under these assumptions we can say that
N(t) changes continuously
Bacteria Growth Revisited
• Upon taking the limit
N(t  t)  N(t) dN

lim
t
dt
t 0
• The continuous model becomes

• Its solution is
dN
 kN
dt
N(t)  N 0e kt

Properties of the Model
• Doubling Time/Half life:
• Steady state
– Ne = 0
dN
0
dt

• Stability
– Ne = 0 is stable if k < 0
– Ne = 0 is 
unstable if k > 0
ln 2
k
Modified Model
• Now assume that growth and
reproduction depends on the available
nutrient concentration
• New Variable
– C(t) = concentration of available nutrient at
time t
• Dimensions of C are mass/vol
Modified Model
• New assumptions
– Population growth rate increases linearly
with nutrient concentration
k(C)  C
  units of nutrient are consumed in
producing
one new unit of bacteria

dC
dN
 
dt
dt

Modified Model
• We now have two equations
dN
 CN
dt
dC
dN
 
dt
dt
• Upon integration we see
C(t)  N(t)
 C0
C
C0

• So any initial nutrient
concentration can only

support a fixed amount

of bacteria
C0


N

The Logistic Growth Model
• Substitute to find
 N 
dN
 rN1 
 K 
dt
dN
  C0  N 
dt
• where
r  C0
Intrinsic
growth rate

K
C0

Environmental
Carrying capacity
The Logistic Growth Model
• Model
• Solution
 N 
dN
 rN1 
 K 
dt
N 0K
N(t) 

N 0  (K  N 0 )ert
• Note: as N K, N/K 1 and 1-N/K
0
• Asthe population size approaches K, the
population growth rate approaches zero
Breakdown
• In general, single species population
growth models can all be written in the
following form
dN
 f (N)  Ng(N)
dt
where g(N) is the actual growth rate.
Actual
growth
 rate
=
Actual birth
rate
Actual death
rate
g(N)  b(N)  d(N)
Breakdown
• The logistic equations makes d(N)
certain assumptions about
the relationship between
population size and the actual
birth and death rates.
 of the
• The actual death rate
d
0
population is assumed to
increase linearly with
population size
d(N)  d0  N

Intrinsic death rate
N
Breakdown
b(N)
• The actual birth rate of
the population is
assumed to decrease

linearly with population
size
b0

b(N)  b0  N
Intrinsic birth rate
b0

N
Breakdown
dN
 g(N)N  b(N)  d(N)N
dt
dN
 b0  N   d0  N N
dt


• Rearrange to get:
    
dN
 b0  d0 N1
N
dt
 b0  d0 
Breakdown
• Now let
Intrinsic
growth rate

r  b0  d0  C0
=
Intrinsic birth
rate
Intrinsic
death rate
b0  d0 C0
K

 

Carrying
capacity
Sensitivity of birth
and death rate to
population size
Plot of Actual Birth and Death Rates

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are needed to see this picture.

K
Assuming Linearity
• Linearity is the simplest way to model the
relationship between population size and
actual birth and death rates
• This may not be the most realistic assumption
for many population
• A curve of some sort is more likely to be
realistic, as the effect of adding individuals
may not be felt until some critical threshold in
resource per individual has been crossed
Solution Profiles
General Single Species
Models
dN
 f (N)  Ng(N)
dt
• Steady States
– Solutions of f(N) = 0
•
N = 0 is always a steady state
• So must determine when g(N) = 0 for nontrivial steady
states
– Example
 N 
dN
 rN1 
 K 
dt
• Steady states are N = 0 and N = K, both always exist.

General Single Species
Models
•
Stability
– How do small perturbations away from
steady state behave?
1. Let N = Ne + n where |n| << 1
2. Substitute into model equation
3. Expand RHS in a Taylor series and
simplify
4. Drop all nonlinear terms
General Single Species
Models
•
Stability
–
Once steps 1 - 4 are preformed, you’ll arrive at
an equation for the behavior of the small
perturbations
dn
 f (N e )n
dt
–

–
n(t) grows if
f (Ne )  0
•
Therefore N = Ne is unstable
•
Therefore
N = Ne is stable

(Ne )  0
N(t) decays if f
n(t)  e f (N e )t
General Single Species
Models
• Stability
– Analysis shows that stability is completely
determined by the slope of the growth
function, f(N), evaluated at the steady
dN
state.
• Example
dt
 N 

dN
 rN1 
 K 
dt
0
K
N
General Single Species
Models
• Stability
 N 
dN
 rN1 
 K 
dt
 N 
f (N)  rN 1 
 K 
 2N 
f (N)  r1

 K 


f (0)  r  0
Ne =
0 is always
unstable
f (K)  r  0
Ne = K is always
stable
Compare Continuous and
Discrete Logistic Model
Discrete
Continuous
dN
 rN
dt
Nt 1  rN t
Solutions grow or decay
-- possible oscillations

 N t 
N t 1  rN t 1 
 K 
Solutions approach N = 0
or N = K or undergo period
doubling bifurcations to
chaos
Solutions grow or decay
--no oscillations

 N 
dN
 rN1 
 K 
dt
All solutions approach N = K

Nondimensionalization
• Definition: Nondimensionalization is an
informed rescaling of the model
equations that replaces dimensional
model variables and parameters with
nondimensional counterparts
Why Nondimensionalize?
• To reduce the number of parameters
• To allow for direct comparison of the
magnitude of parameters
• To identify and exploit the presence of
small/large parameters
• Note: Nondimensionalization is not
unique!!
How to Nondimensionalize
• Perform a dimensional analysis
dN
 rN
dt
Variables/Dimension


N
density
t
time
N(0)  N0
r0
 Parameters/Dimension
r
1/time
N0
density
How to Nondimensionalize
• Introduce an arbitrary scaling of all
variables
N
u
  Bt
A
• Substitute into the model equation
Original Model


dN
 rN
dt
N(0)  N0
Scaled Model
du
AB
 rAu
d
Au(0)  N0
How to Nondimensionalize
• Choose meaning scales
du
AB
 rAu
d
du r
 u
d B
 • Let

Br
A  N0

Au(0)  N0
N0
u(0) 
A
Time is scaled by the
intrinsic growth rate

Population
size is
scaled by the initial size
du
u
d
u(0)  1
How to Nondimensionalize
du
u
d
u(0)  1
• Note: The parameters of the system
are reduced from 2 to 0!!
• There are
no
changes
in
initial
or growth rate that can
 conditions
qualitatively change the behavior of the
solutions-- ie no bifurcations!!
Nondimensionalize the
Logistic Equation